Efron-Stein Inequality proof clarification.

Question

I am currently going through the proof of the Efron-Stein inequality in this set of notes (http://www.econ.upf.edu/~lugosi/mlss_conc.pdf)(P.g. 219-220). However, I have an issue with the final part of the proof where they use the following fact:$$\operatorname{Var}(Y)=\frac{1}{2}E[(Y-Y')^2] \text{, where Y' is an independent copy of Y}$$ to prove that $$E_i[(Z-E_i[Z])^2]=\frac{1}{2}E_i[(Z-Z_i')^2].$$ Can anyone show me how the fact is used in proving the above statement?

Answer 1

Note that $\operatorname{Var}_i(Z) =\frac{1}{2} E_i[(Z-E_i[Z])^2] $. Now $E_i$ is the conditional expectation where all the random variables are given except the $i$-th. Finally note that conditional on $(X_1,\ldots,X_{i-1},X_{i+1},\ldots,X_n)$, $Z$ and $Z_i'$ are independent. Thus the result follows.